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Smoothness of Quotients Associated With a Pair of Commuting Involutions

Published online by Cambridge University Press:  20 November 2018

Aloysius G. Helminck  and
Gerald W. Schwarz
Aloysius G. Helminck
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, U.S.A. e-mail: loek@math.ncsu.edu
Gerald W. Schwarz
Affiliation:
Department of Mathematics, Brandeis University, Waltham, MA 02454-9110, U.S.A. e-mail: schwarz@brandeis.edu
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Abstract

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Let $\sigma $, $\theta $ be commuting involutions of the connected semisimple algebraic group $G$ where $\sigma$, $\theta $ and $G$ are defined over an algebraically closed field $\underset{\scriptscriptstyle-}{k},$ char $\underline{k}$=0. Let $H:={{G}^{\sigma }}$ and $K:={{G}^{\theta }}$ be the fixed point groups. We have an action $\left( H\,\times \,K \right)\,\times \,G\,\to \,G$, where $\left( \left( h,\,k \right),\,g \right)\,\mapsto \,hg{{k}^{-1}},\,h\,\in \,H$ , $k\,\in \,K,g\,\in \,G$. Let $G\,//\,\left( H\,\times \,K \right)$ denote the categorical quotient Spec $\mathcal{O}{{(G)}^{H\times K}}$ . We determine when this quotient is smooth. Our results are a generalization of those of Steinberg [Ste75], Pittie [Pit72] and Richardson [Ric82] in the symmetric case where $\sigma \,=\,\theta $ and $H\,=K$.

Keywords

20G1520G2022E46
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 5 , 01 October 2004 , pp. 945 - 962
Copyright
Copyright © Canadian Mathematical Society 2004

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